We investigate the Cauchy problem for the spin-1 Gross-Pitaevskii(GP)
equation, which is a model instrumental in characterizing the soliton dynamics
within spinor Bose-Einstein condensates. Recently, Geng $etal.$ (Commun. Math.
Phys. 382, 585-611 (2021)) reported the long-time asymptotic result with error
$\mathcal{O}(\frac{\log t}t)$ for the spin-1 GP equation that only exists in
the continuous spectrum. The main purpose of our work is to further generalize
and improve Geng’s work. Compared with the previous work, our asymptotic error
accuracy has been improved from $\mathcal{O}(\frac{\log t}t)$ to
$\mathcal{O}(t^{-3/4})$. More importantly, by establishing two matrix valued
functions, we obtained effective asymptotic errors and successfully constructed
asymptotic analysis of the spin-1 GP equation based on the characteristics of
the spectral problem, including two cases: (i)coexistence of discrete and
continuous spectrum; (ii)only continuous spectrum which considered by Geng’s
work with error $\mathcal{O}(\frac{\log t}t)$. For the case (i), the
corresponding asymptotic approximations can be characterized with an
$N$-soliton as well as an interaction term between soliton solutions and the
dispersion term with diverse residual error order $\mathcal{O}(t^{-3/4})$. For
the case (ii), the corresponding asymptotic approximations can be characterized
with the leading term on the continuous spectrum and the residual error order
$\mathcal{O}(t^{-3/4})$. Finally, our results confirm the soliton resolution
conjecture for the spin-1 GP equation.

Este artículo explora los viajes en el tiempo y sus implicaciones.

Descargar PDF: